Abstract
The betweenness centrality, a path-based global measure of flow, is a static predictor of congestion and load on networks. Here we demonstrate that its statistical distribution is invariant for planar networks, that are used to model many infrastructural and biological systems. Empirical analysis of street networks from 97 cities worldwide, along with simulations of random planar graph models, indicates the observed invariance to be a consequence of a bimodal regime consisting of an underlying tree structure for high betweenness nodes, and a low betweenness regime corresponding to loops providing local path alternatives. Furthermore, the high betweenness nodes display a non-trivial spatial clustering with increasing spatial correlation as a function of the edge-density. Our results suggest that the spatial distribution of betweenness is a more accurate discriminator than its statistics for comparing static congestion patterns and its evolution across cities as demonstrated by analyzing 200 years of street data for Paris.
Highlights
The betweenness centrality, a path-based global measure of flow, is a static predictor of congestion and load on networks
Through simulations of random planar graph models and analytical calculations on Cayley trees, we demonstrate this to be a consequence of a bimodal regime consisting of an underlying tree structure for high betweenness centrality (BC) nodes, and a low BC regime corresponding to loops providing local path alternatives
One sees a dramatic difference at the scale of three thousand squarekilometers (Supplementary Figs. 1c, d) where we observe that the BC distribution for cities within each category is virtually identical, and bimodal, with two regimes separated by a bump roughly at gB∼N
Summary
The betweenness centrality, a path-based global measure of flow, is a static predictor of congestion and load on networks. Empirical analysis of street networks from 97 cities worldwide, along with simulations of random planar graph models, indicates the observed invariance to be a consequence of a bimodal regime consisting of an underlying tree structure for high betweenness nodes, and a low betweenness regime corresponding to loops providing local path alternatives. There have been limited studies of the BC distribution in its entirety, with the majority of analyses instead focusing on the average BC (proportional to the average shortest path44) or on its maximum value[45,46] To fill this gap in our understanding of this important class of networks, we conduct here a large-scale empirical study of the BC across 97 of the world’s largest cities as measured by population (details on dataset in Methods). For the case of street networks, as long as planarity is conserved, bottlenecks continue to persist, and the effect of planned interventions to alleviate structural congestion will be limited primarily to load redistribution, a feature confirmed by analyzing 200 years of data for central Paris
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